Approximation in higher-order Sobolev spaces and Hodge systems

Let $d\ge 2$ be an integer, $1\le l\le d?1$ and φ be a differential l-form on ${\mathbb{R}}^d$ with ${\dot{W}}^{1,d}$ coefficients. It was proved by Bourgain and Brezis ([5, Theorem 5]) that there exists a differential l-form ψ on ${\mathbb{R}}^d$ with coefficients in $L^{\infty }\cap {\dot{W}}^{1,d}$ such that $d\phi =d\psi$. In the same work, Bourgain and Brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order space ${\dot{W}}^{2,d/2}$ or more generally in the fractional Sobolev spaces ${\dot{W}}^{s,p}$ with $sp=d$. We give a positive answer to this question, provided that $d?\kappa \le l\le d?1$, where κ is the largest positive integer such that $\kappa <\min ?(p,d)$. The proof relies on an approximation result (interesting in its own right) for functions in ${\dot{W}}^{s,p}$ by functions in ${\dot{W}}^{s,p}\cap L^{\infty }$, even though ${\dot{W}}^{s,p}$ does not embed into $L^{\infty }$ in this critical case. The proofs rely on some techniques due to Bourgain and Brezis but the context of higher order and/or fractional Sobolev spaces creates various difficulties and requires new ideas and methods.     

Sobolev spaces and the Cayley transform

The generalised Cayley transform  from an Iwasawa -group into the corresponding real unit sphere  induces isomorphisms between suitable Sobolev spaces and . We study the differential of  , and we obtain a criterion for a function to be in  .

     

Approximation by polynomials and smooth functions in Sobolev spaces with respect to measures

The density of polynomials is straightforward to prove in Sobolev spaces W^k,p((a,b)), but there exist only partial results in weighted Sobolev spaces; here we improve some of these theorems. The situation is more complicated in infinite intervals, even for weighted L^p spaces; besides, in the present paper we have proved some other results for weighted Sobolev spaces in infinite intervals.     

On the periodic KdV equation in weighted Sobolev spaces

We prove well-posedness results for the initial value problem of the periodic KdV equation as well as Kam type results in classes of high regularity solutions. More precisely, we consider the problem in weighted Sobolev spaces, which comprise classical Sobolev spaces, Gevrey spaces, and analytic spaces. We show that the initial value problem is well posed in all spaces with subexponential decay of Fourier coefficients, and ‘almost well posed’ in spaces with exponential decay of Fourier coefficients.     

On flows associated to Sobolev vector fields in Wiener spaces: An approach à la DiPerna-Lions

In this paper we extend the DiPerna-Lions theory of flows associated to Sobolev vector fields to the case of Cameron-Martin-valued vector fields in Wiener spaces E having a Sobolev regularity. The proof is based on the analysis of the continuity equation in E, and on uniform (Gaussian) commutator estimates in finite-dimensional spaces.     

Notes on limits of Sobolev spaces and the continuity of interpolation scales

We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz'ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also give a new approach to the problem of how to recognize constant functions via Sobolev conditions.

     

Continuity of the maximal operator in Sobolev spaces

We establish the continuity of the Hardy-Littlewood maximal operator on Sobolev spaces , . As an auxiliary tool we prove an explicit formula for the derivative of the maximal function.

     

The IVP for the Benjamin-Ono equation in weighted Sobolev spaces

We study the initial value problem associated to the Benjamin-Ono equation. The aim is to establish persistence properties of the solution flow in the weighted Sobolev spaces , s∈R, s?1 and s?r. We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demonstrate that our persistence properties are sharp.     

Further spectral properties of the weighted finite Fourier transform operator and approximation in weighted Sobolev spaces

In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWF). This set of special functions have been defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super‐exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.     

Flat wavelet bases adapted to the homogeneous Sobolev spaces

We present a construction of “flat wavelet bases” adapted to the homogeneous Sobolev spaces ?^s (?ⁿ ). They solve the problem of the phenomenon of infrared divergence which appears for usual wavelet expansions in ?^s (?ⁿ ): these bases remove the divergence in the case s – n /2 ? ? since they are also bases of the realization of ?^s (?ⁿ ). In the critical case s – n /2 ∈ ?, they provide a confinement of the divergence in a “small” space. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cubature over the sphere in Sobolev spaces of arbitrary order

This paper studies numerical integration (or cubature) over the unit sphere for functions in arbitrary Sobolev spaces H^s(S²), s>1. We discuss sequences of cubature rules, where (i) the rule Q_m(n) uses m(n) points and is assumed to integrate exactly all (spherical) polynomials of degree ≤n and (ii) the sequence (Q_m(n)) satisfies a certain local regularity property. This local regularity property is automatically satisfied if each Q_m(n) has positive weights. It is shown that for functions in the unit ball of the Sobolev space H^s(S²), s>1, the worst-case cubature error has the order of convergence O(n^-s), a result previously known only for the particular case . The crucial step in the extension to general s>1 is a novel representation of , where P_ℓ is the Legendre polynomial of degree ℓ, in which the dominant term is a polynomial of degree n, which is therefore integrated exactly by the rule Q_m(n). The order of convergence O(n^-s) is optimal for sequences (Q_m(n)) of cubature rules with properties (i) and (ii) if Q_m(n) uses m(n)=O(n²) points.     

Characterization of Sobolev and BV spaces

The main results of this paper are new characterizations of W1,p(Ω), 1<p<∞, and BV(Ω) for Ω⊂RN an arbitrary open set. Using these results, we answer some open questions of Brezis (2002) [10] and Ponce (2004) [25].     

Estimates of difference norms for functions in anisotropic Sobolev spaces

We investigate the spaces of functions on ?ⁿ for which the generalized partial derivatives Dequation/tex2gif-sup-2.gif_kf exist and belong to different Lorentz spaces Lequation/tex2gif-sup-3.gif . For the functions in these spaces, the sharp estimates of the Besov type norms are found. The methods used in the paper are based on estimates of non‐increasing rearrangements. These methods enable us to cover also the case when some of the p_k's are equal to 1. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Summing Property of the Sobolev Embedding Operators

We prove that the Sobolev embedding operator S _d,k,p : , where 1/s=1/p-k/d , is (v,1) -absolutely summing for appropriate v > 1 . The result is optimal for s 2 .     

Orthogonal systems of spline wavelets as unconditional bases in Sobolev spaces

We exhibit the necessary range for which functions in the Sobolev spaces $L_p^s$ can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle–Lemarié wavelets. We also consider the natural extensions to Triebel–Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich, where analogous results were established for the Haar wavelet system.     

A proof of the trace theorem of Sobolev spaces on Lipschitz domains

A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on . It is proved that the trace operator is a linear bounded operator from to for .